A373361 -id:A373361

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A276085

Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

+0
168

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Completely additive with a(p^e) = e * A002110(A000720(p)-1).

This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022

On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..2310

Index entries for sequences related to primorial base

Index entries for sequences related to primorial numbers

FORMULA

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).

a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.

Other identities.

For all n >= 0:

a(A276086(n)) = n.

a(A000040(1+n)) = A002110(n).

a(A002110(1+n)) = A143293(n).

From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)

a(A283477(n)) = A283985(n).

a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]

When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:

a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).

a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).

In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:

a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).

a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).

a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).

a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).

The sum or difference of the rhs-sequences is A108951:

a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).

a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).

Here the two sequences are inverse permutations of each other:

a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).

a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).

a(A346101(n)) = A289234(n). [Self-inverse]

Other correspondences:

a(A324350(x,y)) = A324351(x,y).

a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]

a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]

(End)

MATHEMATICA

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)

f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

PROG

(Scheme, with memoization-macro definec)

(definec (A276085 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A002110 (+ -1 (A055396 n)))) (A276085 (A028234 n))))))

(PARI) A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

(Python)

from sympy import primorial, primepi, factorint

def a002110(n):

return 1 if n<1 else primorial(n)

def a(n):

f=factorint(n)

return sum(f[i]*a002110(primepi(i) - 1) for i in f)

print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

CROSSREFS

A left inverse of A276086.

Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).

Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.

Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).

Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].

Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].

Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).

Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 21 2016

EXTENSIONS

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024

STATUS

approved

A373368

a(n) = gcd(n, A059975(n)), where A059975 is fully additive with a(p) = p-1.

+0
6

1, 1, 1, 2, 1, 3, 1, 1, 1, 5, 1, 4, 1, 7, 3, 4, 1, 1, 1, 2, 1, 11, 1, 1, 1, 13, 3, 4, 1, 1, 1, 1, 3, 17, 5, 6, 1, 19, 1, 1, 1, 3, 1, 4, 1, 23, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 29, 1, 4, 1, 31, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 37, 5, 4, 1, 3, 1, 8, 1, 41, 1, 2, 5, 43, 3, 1, 1, 9, 1, 4, 1, 47, 1, 1, 1, 1, 1, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };

A373368(n) = gcd(n, A059975(n));

CROSSREFS

Cf. A059975, A108269 (positions of even terms), A359794 (of odd terms), A359832 (parity of terms).

Cf. also A082299, A373361, A373369.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2024

STATUS

approved

A373362

a(n) = gcd(A001414(n), A276085(n)), where A001414 is the sum of prime factors with repetition, and A276085 is the primorial base log-function.

+0
11

0, 1, 1, 2, 1, 1, 1, 3, 2, 7, 1, 1, 1, 1, 8, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 12, 2, 1, 1, 8, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 4, 17, 1, 1, 8, 1, 2, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 9, 1, 1, 4, 1, 1, 2, 2, 1, 8, 1, 1, 1, 20, 1, 2, 1, 12, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

As A001414 and A276085 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication: For example, A373373, which gives the indices of multiples of 3.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

PROG

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

A002110(n) = prod(i=1, n, prime(i));

A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };

A373362(n) = gcd(A001414(n), A276085(n));

CROSSREFS

Cf. A001414, A002110, A276085.

Cf. A345452 (positions of even terms), A373373 (positions of multiples of 3).

Cf. also A082299, A373361, A373363, A373367.